A mathematical model and computer program of economic evaluation and prediction

A

Chen Zheng Agricultural Information

Bank of China, Guangxi Branch and Computer Department

27 Tiantao Rd. Nanning, Guangxi,

PR China 530022

ABSTRACT In this article, predicted distribution

regularity,

Model of Specialists computer

1.

value and measured

Based

Prediction

value are suppoed to have the same

on this supposition, and give relevant

we put fonvard concepts,

deductive

a Mathematics proofs,

and a

program.

INTRODUCTION

In the past, people generally adopted mathematical methods of regression analysis, system dynamics, and input-output, etc. to predict economic targets. In the process of prediction, these methods depend a great deal on some fured tendencies and data. But when some economic targets, such as investment, project rate of return, and shipping insurance, etc., the tendencies are not clear and part of the data are not collected in prediction. In general, solving this kind of problem adopts a manmade analysis. First, we use individual analysis to determine the solution. This method has a larger randomness and is affected by everyone. Secondly, we use a specialists collective to determine the solution. This method can reduce the mistakes caused by an individual qualitative analysis. But this method also has some problems, such as the difficulty of coordinating when different specialists’ predicted values are more different. This is because this kind of prediction depends more on expressive ability in all respects. At the same time, its efficiency is not high. In order to solve the problem, we put forward the Mathematics Model of Specialists Prediction. APPLIED MATHEMATZCS AND COMPUTATION 59:63-72 0 Elsevier Science Publishing Co., Inc., 1993 655 Avenue of the Americas, New York, NY 10010

(1993)

63 0096.3003/93/$6.00

64 2.

C. ZHENG MODELS

BASIC

HYPOTHESIS

Suppose the predicted process and measured process have the same model. Now we analyze as follows. The measurement’s essential factors can be summarized as objects to measure, such as measured centigrade, etc., circumstances to measure, such as atmospheric temperature, etc., means to measure, such as tools and methods to measure, man’s experience, and engineering level, etc. Similarly, when measuring or calculating some economic

targets

statistically,

we can

summarize

three

essential

factors:

(1)

targets to calculate such as internal rate of return, (2) basic circumstance conditions to calculate such as data precision and data range, (3) means to calculate such as calculating methods and treating experience, etc. Therefore, we think statistical calculation is also a kind of measurement. If the known data range is narrow, precision is poor. It is called estimated data. In general, estimated data and evaluation-prediction have the same precision grade. It is difficult to analyze rough data and to calculate known data. Prediction is the ability to analyze and calculate situations in the future and to determine some data targets. The former is standing nowadays and the latter is standing in the future. Evaluation and prediction value’s probabilities are larger near statistical calculation of truth value (precision value). In the objective world, measurements satisfy the normal distribution rule. For example, we measure an object’s diameter twenty times. The measurement data satisfies the normal distribution. According to the above analysis, we think prediction satisfies the normal distribution. 3. 3.1.

MODEL

AND

DEFINITIONS

Formula of model

F(t.n)

3.2.

Definition

= 0.5 + sign(t)

X 11(0.5,

n/2)

f

2

(I)

of model

sign(t)

=

+1 i -1

tao t
ZJO.5, c + 1) = ZJO.5, c) + 0.5U,(O.5, UJO.5, c + 1) = qo.5,

c)

c) * (0.5 + c) * (1 - x)/c

(2)

Economic

Evaluation

When

When

n =

65

and Prediction

Zk (k = I,?,, . . . , n), the initial value of (2) is

Z,(O.5,1)

= A

U,(O.5,1)

= 0.5 * &- * (1 - X).

n = Zk + 1 (k = 0, 1, . . . , n>, the initial value of (2) is

Z,(O.5,0.5)

= 1 - 2. tg-‘(d-)/T

U*(O.5,0.5)

= 4-/7r,

and c = n/2,

x = t’/(n

+ t’),

where (1)

ai is the p re d’rc t’ion value given by specialists and (i = 1,2, . . . , n> and of specialists; a is the expectation value of an economic target given by users; a, is the average of a, (i = 1,2,. . . , n); s is the sample standard deviation of ai using s to express risk; t is the integral upper limit of probability density; and F(t, n) is the probability value of a using F(t, n) to express credence.

n are the number

(2) (3) (4) (5) (6)

66 4.

C. ZHENG APPLIED

EXAMPLE

WuZhou agriculture bank needs to predict the deposied amount on June 1989. At first, let twelve specialists evaluate and predict and give the amount deposited. Then put forward the expected ment as shown in Table 1.

value a = 14, by planning

depart-

Put twelve known data of predicted value and expectation value into the computer. After calculating the screen shows the results: The credence of a = 13.30 is 0.89196; the standard deviation is 0.3672. Adjust a to 13.28, then put it into the computer. After calculating the screen shows the results: credence of a = 13.28 is 0.91682; the standard deviation is 0.3672. The meaning of the 91.69%,

which

calculated

is the amount

results

above

deposited:

1.328

express billion

the

credence

yan. The

of

credence

shows specialists’ views of expectation value to hold true. Using the sample standard deviation to express risk, which is determined by twelve predicted values, shows the dispersed circumstances among specialists data. The less risk the better. The last statistical deposited amount of June 1989 is 1.32768 billion. The relative value is 0.02%.

5.

CONCLUDING

error

between

value and the

statistical

REMARKS

The theory of the Mathematics and easy to treat

the predicted

by computer.

Model

of Specialist

Prediction

It is a kind of prediction

model

is reliable to solve

individual randomness and overall regularity. It is a supplement to traditional prediction methods. It cannot only be used in predicting economic targets, but also in predicting other targets. In economic activity, it is an attempt to determine regularity as possible in undetermined regularity. According to the model, we draw up a computer program to calculate and predict the internal rate of return and cost rate of return in Appendix 2. In the process of using it, relevant targets of internal rate of return and cost rate of return and twelve specialists forecast values are inputted. The program can calculate credence and standard deviation of the expectation value relevant targets.

TABLE 1 al 13.94

a2 13.46

a3 12.89

a4 13.44

a5 13.27

a6 13.19

a7 13.5

a8 12.9

a9 12.78

al0 13.97

all 13.8

al2 13.3

67

Economic Evaluation and Prediction APPENDIX

1: MODEL

DEDUCTIVE

Hypothesis According

to basic hypothesis,

PROOF

the randon variables

a, (i = 1,2, . . . , n) are

independent of each other and obey the same normal distribution N( a,, u ‘>. When we replace the total standard deviation u b the sample standard deviation s, the random variable t = (a, - a)/(~/ P n > obeys t distribution, whose degree of freedom is 12. Since this distribution does not relate to the total standard deviation, the credice of a can be estimated under no known standard deviation.

PROOF. The probability

f(t) =

density function

W” + W2) &&I(

n/2)

This is the density random variable t.

-(n+l)/Z * G+

function,

whose degree

According

&f(t) =

and B(m,

to (4), translating

&B(

1

l/2,

of freedom

is n followed

(3)

by

I-(m) *r(n)

qm + n)

n> function

(4)

are gamma and beta,

(3) into the B function

-(n+l)/Z n/2)

is as follows:

--co < t < +w.

t2’n)

B(m, n) =

The names of r(m) tively.

of t distribution

form,

---cc)
. (’ + t2’n)

respec-

< +m.

(5)

Now through the recurrence formula of B distribution, we can see the recurrence formula of t distribution. The probability density function of b distribution is

.fB(X~~>C) =

B(:,c) .x-l.

(1

_

q-1

(0 < X < 1, U > 0, c > 0).

(6)

C. ZHENG

68 The distribution

of B distribution

function

L(U>c>= p

B(;

is

.xu-‘.

(1

-

x)“-‘&

c)

(0 < x < 1, u > 0, c > 0).

U,(U,c) = then the recurrrence

B(;,c) .XU.(l

formula of B distribution

-x>“,

(7)

(8)

is

Z,(U + 1, c) = Z,(U, c) - U,(U, c)/U

Z,(U, c + 1) = Z,(U, c) + U,(U, c)/c u,(u+ U,(U,c+

1,c)

= u,(u,c)~(u+c)~x/u

1) = u,(u,c)qU+c)*(l

-x)/c.

The recurrence formula of t distribution formula of B distribution. Let

U = l/2,

c =n/2,

x = -

(9)

can be let by the recurrence

t2

n + t2’

dx =

2tn dt

(n + t2)2 ’

and inset (61, then

_fB(X> 112,n/2) =

=

H(1/:. (_YJ’^‘. (-2_)n’z-1 .

n,2)

1

jn

i'k3(1,'2,n,'2)

zt2 + l)-"")'2. (n :tI')?.

Economic Evaluation and Prediction

After arranging

and differentiating

69

both sides, -(n + 1)/Z

.2 dt

= zf(t) The function

F(t,

According

of t distribution

n) = /f

f( X, n)

--m

dt = 2f(t,

n) = l/2

(10)

is

dx = /’

to (lo), inset the equation

F(t,

n) dt .

f( X, n) d3c + I,tf( --m

X, n) dx.

as above; then

+ 1,‘2/“fk(

X, l/2,

n/2)

dx.

0

According

to (7) and the character

of the integral,

F( t, n) = 0.5 + sign(t)

. Ix( l/2,

n/2)

+ 2.

It is the same with (1).

APPENDIX

2: COMPUTER

PROGRAM

/ * program CZ.c # include (stdi0.h) # include (math.h) # include (time.h) # include (decima1.h) maim ) { double powi >, sqrt( >; int n, i, n2, sign; float ABSl, il, i2, i3, i4, ii, sslf, ss2f, xlp, x2p, y, x2es, xles, IRR, BCR, u, p, tt; float xx1[20], xx2[20], ri, zhong, al, bl, cl, dl, tl, t2,x, pl, p2; / c __________ enter &a __________* / printff“\033[2J\033[1; 1H”); printf(“\n please enter IRR = \t”); scanf(“%f’, & IRR);

70

C. ZHENG printf(“\n scanf(“%f’,

please enter BCR

= \t”);

& BCR);

printf(“\n

please enter maximum

scan(“%f’,

& il);

printf(“\n scanf(“%f printf(“\n scanf(“%f printf(“\n

please enter maximum

increment

of IRR

= \t”);

increment

of BCR

increment

of IRR = \t”);

increment

of BCR

= \t”);

‘, & i2); please enter maximum ‘, & i3); please enter maximum

/ * -------- ri is random

number.

--------

= \t”);

* /

/ * ----- xx l[‘]1 an d xx 2[.]1 are 12 values of IRR and, BCR, respectively. for (i = 1; i < = 12; i + + >

----- * /

xlp = xlp + xxl[i]; x2p = x2p + xx2[i]; 1 xlp = xlp/12; x2p = x2p/12; / * ----- here, weighting is equal to everybody.

-----

*/

/ * --- xlp and x2p are average values of IRR and BCR, respectively.

--- * /

Economic

Evaluation

and Prediction

71

for (i = 1; i < = 12; i + + ) I sslf = (xxl[i] - xlp)*(xxl[i] ss2f = (xx2[i] - x2p)*(xx2[i] ] sslf = sslf/ll; ss2f = ss2f/ll; / * -- sslf and ss2f are standard

- xlp) + sslf; - x2p) + ss2f;

deviation

of IRR and BCR, respectively.

*/ if (sslf < 0.0000001) xles = 9; / * giving xles a big enough else

number.

--

*/

zhong = sslf/l2; xles = (xlp - IRR)/sqrt(zhong); ] if (ss2f < 0.0000001) x2es = 9; / * giving x2es a big enough number. * /. else { zhong = ss2f/12; x2es = (x2p - BCR)/sqr&hong); ] / * -- xles and x2es are probability density fo IRR and BCR, respectively. -- * / tl = xles; t2 = x2es; if (tl > = 0) sign = 1; else sign = -1; tt = tl*tl; x = tt/(12 + tt); / * -- x is probability density ingegral upper value of T distribution. -- * / p = sqrttx); u = p*(l - x)/2; n2 = 10; / * n2 represent freedom N - 2 * / for (i = 2; i < n2; i = i + 2) { p = p + 2*u/i; u = u*(l + i)*(l - x)/i; ] pl = 0.5 + sign*pp/2; / * pl is probability of IRR * / / * _____ _____________________________ ______* / if (t2 > = 0) sign = 1;

72

C. ZHENG

else sign = -1; tt = t2*t2; x = tt/(12 + tt); / * -- x is probability density integral upper value of T distribution. -- * / p = sqdx); u = p*(l - x)/2; n2 = 10; / * n2 represent freedom N - 2 * / for (i = 2; i < n2; i = i + 2) I p = p + 2*u/i; u = u*(l + i)*(l - x)/i; 1 p2 = 0.5 + sign * p/2; / * p2 is probability of BCR * / sslf = sqr&slf); ss2f = sqrt(ss2D; ABSl = 100 * (IRR - ii) * pi * (BCR * p2) * (BcR * ~2); / * --- ABSl is synthetic evaluate value. --- * / printr‘\nIRR of project = %f probability of IRR = %f risk of IRR = %f’, IRR, pl, sslfl; printfT“\nBCR of project = %f probability of BCR = %f risk of BCR = %f’, BCR, p2, ss2f); printfI“\n the synthetic evaluate value of IRR and BCR%f’, ABSl); 1 / *